A formulation of sum rules expected at finite temperatures was first obtained in [1]} and more recently also in [2]}, [3]}.
Experimentally, the in-medium spectral function of the \(\rho \)  meson is accessible by dilepton spectroscopy.
The results of various heavy ion experiments, that will be presented in Sec. REF , provide convincing evidence for its substantial modifications in dense and hot nuclear matter.
Hence, low-mass dilepton data provide a valuable constraints for theoretical models.
Up to now, no complementary experimental information on the in-medium \(a_1\)  spectral function is available.
Nevertheless, the detailed understanding of the vector meson spectral function alone together with constraints imposed by the WSR and QCDSR sum rules provide already very valuable insight to chiral restoration effects on hadronic spectra.
This has been demonstrated by Hohler and Rapp [3]} in an analysis using the spectral function of the \(\rho \)  meson calculated with hadronic models [5]}, [6]}, described in more details in the next section.
The calculations allows to establish the full evolution of the vector spectral function as function of the temperature and the baryo-chemical potential.
Using these spectral functions and a temperature dependence of the condensates as defined in the right hand side of Eq. REF , obtained from thermal lattice QCD calculation for \(\mu _b=0\) , and assuming otherwise a hadron gas equation-of-state as the input, QCDSR were calculated for vanishing chemical potential.
The result shows that the sum rules are satisfied within few percent, which is already a non-trivial result.
In the next step, the evolution of the axial vector spectral function has been calculated requesting conservation of the QCDSR and WSR.
The axial spectral function was modeled in the calculations by \(a_1\)  with a Breit-Wigner distribution with variable peak position and a width and a continuum contributions representing vector-axial vector mixing.
The calculation demonstrate, see Fig. REF , gradual reduction of the \(a_1\)  mass and an increasing width of the Breit-Wigner peak, while the \(\rho \)  mass stays nearly constant but the width is increasing as well with a shoulder appearing on the low-mass side of the distribution (see for details fig.3 in [3]}).
The obtained behavior shows that the spectral functions of vector-axial chiral partners become degenerate at \(T=T_{\mathrm {pc}}\)  , as expected for the chiral symmetry restoration.
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